In this problem we will look at properties of solids with base on the x-y plane

Question

In this problem we will look at properties of solids with base on the x-y plane given by the circle x^2 + y^2 = a^2 and its interior, where a > 0 is to be given in parts (i) and (ii) below. (i) Let a = 2, and assume that cross-sections of the solid perpendicular to the x-axis are circular disks with diameters lying in the x-y plane. Find the area A(x) of the cross-section at location x. See picture here. A(x)= (ii) Let a = 6, and assume that cross-sections of the solid perpendicular to the x-axis are equilateral triangles with bases in the x-y plane. Find the area A(x) of the cross-section at location .x. See picture here. A(x) = (iii) Find the volume of the solid described in (i) above. V =

Explanation / Answer

1. At a distance x from Origin, the radius of the cross section will be sqrt (4-x^2), So the are of cross section = 3.14* (4-x^2)

2. At a distance x, the area of equilateral triangle cross section will be (sqrt 3)/4*2*(36-x^2)

3. Volume = Area* height= int(-x to x) (3.14*(4-x^2))dx= 25.12 x- 2.09x^3

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